A new proof of the existence of hierarchies of Poisson-Nijenhuis structures
type: Item
instance of: scholarly article
title: A new proof of the existence of hierarchies of Poisson-Nijenhuis structures
zbMATH Open document ID: 1086.53098
author: Juan Monterde
published in: Portugaliae Mathematica. Nova Série
publication date: 2005-03-30T00:00:00Z
full work available at URL: https://eudml.org/doc/52320
review text: The paper under review provides a new approach to the phenomenon of hierarchies of Poisson-Nijenhuis structures in the sense of the paper by \textit{Y.~Kosmann-Schwarzbach} and \textit{F.~Magri} [Ann. Inst. Henri Poincaré, Phys. Théor. 53, No. 1, 35--81 (1990; Zbl 0707.58048)]. Thus, let \(P\colon T^*M\to TM\) be a Poisson tensor field and \(N\colon TM\to TM\) a Nijenhuis tensor field on the manifold \(M\). For every \(t\in{\mathbb R}\) define \(\Phi_t=\exp(tN)\colon TM\to TM\). One of the key statements (Proposition~3) of the paper says that the pair \((P,N)\) is a Poisson-Nijenhuis structure if and only if so are all the pairs \((P,\Phi_t)\) whenever \(t\in{\mathbb R}\). And one of the consequences of this fact (Corollary~3) is that, if \((P,N)\) is a Poisson-Nijenhuis structure then the pair \((\Phi_sP,\Phi_t)\) is a Poisson-Nijenhuis structure and moreover \(\Phi_sP+\Phi_tP\) is a Poisson tensor field for arbitrary \(s,t\in{\mathbb R}\). Under the same hypothesis, for all \(k,l\in{\mathbb N}\), the pair \((N^kP,N^l)\) is a Poisson-Nijenhuis structure and \(N^kP+N^lP\) is a Poisson tensor field (Corollary~5). An illustrating example concludes this well-written paper.
reviewed by: Daniel Beltiţă
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 2149496
zbMATH Keywords: Poisson-Nijenhuis structure
zbMATH Keywords: Gerstenhaber algebra
zbMATH Keywords: infinitesimal generator
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2149496